Of course, the trouble is getting heavy stuff to drop in. Planets in
the vicinity suddenly look tempting...

"Hi! We're your soon-to-be neighbours, the Sandberg Noosphere V 14.1,
and we thought we ought to drop in. You see, your sun is going nova in
a few million yeas, and we think it would be uncomfortably close to us
by then. I hope you don't mind if we shorten that span a bit by
dropping your planets into the sun, especially the terrestrial ones -
we do hope you have plenty of iron. After all, they will be gone
anyway soon. You can always move to our place, we have room for
another layer on our Dyson sphere."

Seriously, I think the rate of material exchange in big stars is
likely too low to make it easy to just drop planets into them. The
material will likely just drift around in the upper layers. Besides,
it probably takes a lot of matter to age the star seriously. Still, it
might be worth checking out the physics of lithium stars, which
appears to have swallowed a few planets.

> Dropping a> black hole quickly through the center of a star will make it YOUNGER, because> more of the heavy elements will be at the core, so when the black hole passes> through the star it will eat a tunnel through the star, sucking up a greater> proportion of the heavy element mass.

It depends a bit on how much the hole manages to suck up; a small
black hole will likely not eat a star very quickly. On the other hand,
the luminosity of the hole will be quite high, and it will heat up the
core a bit. Most likely the hard radiation from the hole will break up
nuclei too. So it is not obvious in what direction the hole will move
the core.

One idea that hit me recently: add pairs of small black holes and
allow them to coalesce. The result is a strong burst of gravitational
radiation that likely could mix the star a bit. So you add small black
holes on carefully calculated orbits, allow them to eat up some of the
core, break down heavy nuclei and heat it, and then they coalesce,
creating a stronger mixing between the hot and heavy core and the
lighter stuff around it. This could likely make sure the star remains
fusing for a while longer - much less risk of iron collapse. Of
course, eventually there is going to be a collapse as the black hole
eats the star, but at that point you are far away...